I will describe how quantum properties of black holes can be used to
learn about the short-distance structure of quantum gravity. Much of
earlier work on this topic has been in the limit of of large area of
the black hole. Recent progress has made it possible to study finite
size quantum effects systematically in some examples. These results
have implications for studying the AdS/CFT holographic correspondence
at the quantum level at finite N and have revealed a fascinating
connection with the mathematics of mock modular forms.
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